Hint.

Recall that possibilities for the solution set of a system of linear equation is either no solution (inconsistent) or one unique solution or infinitely many solutions.

A homogeneous system is a system with zero constant terms.
A homogeneous system always has the zero solution.

Solution.

(a) Note that by the given condition $a/d=b/e=c/f$, the numbers $b, e, c$ are not zero. The augmented matrix of the system is
$\left[\begin{array}{rr|r}
a & b & c \\
d & e & f
\end{array}\right]$.
We apply the elementary row operations as follows.

(b) The system is homogeneous, thus it has the zero solution. Since the coefficient matrix $A$ is singular, the system has non-zero solution as well.
Therefore, the only possibility is that the system has infinitely many solutions.

(c) A homogeneous system has the zero solution hence it is consistent. Since there are more unknowns than equations, the system must have infinitely many solutions.

(d) The last row of the reduced row echelon form matrix of the augmented matrix is $[000|1]$.
It corresponds to the equation
\[0x_1+0x_2+0x_3=1.\]
Equivalently, this is $0=1$ and this is impossible. Thus the system has no solution (an inconsistent system).

Solving a System of Linear Equations By Using an Inverse Matrix
Consider the system of linear equations
\begin{align*}
x_1&= 2, \\
-2x_1 + x_2 &= 3, \\
5x_1-4x_2 +x_3 &= 2
\end{align*}
(a) Find the coefficient matrix and its inverse matrix.
(b) Using the inverse matrix, solve the system of linear equations.
(The Ohio […]

Linear Algebra Midterm 1 at the Ohio State University (1/3)
The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017.
There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold).
The time limit was 55 minutes.
This post is Part 1 and contains the […]